PSFC/JA-14-24 Synergistic Cross-Scale Coupling of Turbulence in a Tokamak Plasma N.T. Howard, 1 C. Holland, 2 A.E. White, 3 M. Greenwald, 3 and J. Candy 4 1 Oak Ridge Institute for Science and Education (ORISE), Oak Ridge, Tennessee 37831, USA 2 University of California - San Diego, La Jolla, CA, 92093 3 MIT - Plasma Science and Fusion Center, Cambridge, MA, 02139 4 General Atomics, PO Box 85608, San Diego, CA 92186 July, 2014 Plasma Science and Fusion Center Massachusetts Institute of Technology Cambridge MA 02139 USA This work was supported by the U.S. Department of Energy, Grant No. DE-AC02-05CH11231 and DE-FC02-99ER54512-CMOD and in part by an appointment to the US DOE Fusion Energy Postdoctoral Research Program administered by ORISE. Reproduction, translation, publication, use and disposal, in whole or in part, by or for the United States government is permitted.
Synergistic Cross-Scale Coupling of Turbulence in a Tokamak Plasma N.T. Howard, 1, a) C. Holland, 2 A.E. White, 3 M. Greenwald, 3 and J. Candy 4 1) Oak Ridge Institute for Science and Education (ORISE), Oak Ridge, Tennessee 37831, USA 2) University of California - San Diego, La Jolla, CA, 92093 3) MIT - Plasma Science and Fusion Center, Cambridge, MA, 02139 4) General Atomics, PO Box 85608, San Diego, CA 92186 (Dated: 30 July 2014) For the first time, realistic, nonlinear gyrokinetic simulations spanning both the ion and electron spatio-temporal scales have demonstrated the coexistence of ion (k θ ρ s O(1.0)) and electron-scale (k θ ρ e O(1.0)) turbulence in the core of a tokamak plasma and the possible explanation of a documented discrepancy between simulation and experiment. These cutting-edge, multi-scale simulations utilized the GYRO code [J. Candy and R.E. Waltz, J Comput. Phys. 186, 545(2003)]tostudythecoupling of ion and electron-scale turbulence in the core (r/a = 0.6) of an Alcator C-Mod L-mode discharge shown previously to exhibit an under-prediction of the electron heat flux when using simulations only including ion-scale turbulence. Electron-scale turbulence is found to play a dominant role in setting the electron heat flux level and radially-elongated streamers are found to coexist with larger ion-scale eddies in experimental plasma conditions. Inclusion of electron-scale turbulence in these simulations is found to increase both ion and electron heat flux levels by enhancing the transport at the ion-scale while also driving electron heat flux at sub-ρ i scales. These results suggest that previously observed discrepancies in the electron heat flux may be explained by inclusion of electron-scale contributions in gyrokinetic simulation. a) Electronic mail: nthoward@psfc.mit.edu 1
I. INTRODUCTION Despite decades of research, anomalous electron heat transport in fusion plasmas is still not well understood and is extremely important to the success of ITER and future fusion reactors. It is generally believed that most anomalous transport observed in magnetic fusion devices is the result of long-wavelength (k θ ρ s < 1.0), drift-wave-type instabilities driven unstable by free energy in the plasma s background gradients 1. Here k θ is the perpendicular wavenumber, ρ s = c s /Ω ci is the sound speed gyroradius, Ω ci = eb/m i c is the ion gyrofrequency, and c s = T e /m i is the sound speed. Due to their large eddy size and low frequency, the most virulent of these instabilities are identified as Ion Temperature Gradient (ITG) and Trapped Electron Modes (TEM). However, recent experimental evidence 234 suggests this paradigm may not be sufficient for describing electron transport in the tokamak core. Using gyrokinetic simulations focused on electron-scale turbulence, Dorland et al. demonstrated that short wavelength electron temperature gradient (ETG) driven turbulence could form radially elongated structures known as ETG streamers 5 that are capable of driving experimental levels of electron heat transport 6. However, the actual role of streamers has been unclear, as the presence of ion-scale turbulence has been predicted to shear these structures apart, destroying their impact on the electron heat transport 7. As a result, ETG is generally thought to be significant in plasma conditions where the role of ion-scale turbulence is limited, i.e. by ExB shear suppression 8,9,electroninternaltransportbarriers 10, or in conditions similar to the H-mode pedestal 11,leavingtheoriginofelectrontransport under many more standard conditions not well understood. Investigation of high-k (k θ ρ s > 1.0) ETG turbulence using gyrokinetic simulation has, to date, focused predominantly on electron-scale simulation 5,6,9,10,12. In conditions with unstable low-k (k θ ρ s < 1.0) turbulence, such simulations do not provide an accurate representation of reality, as they do not include long wavelength, ion-scale dynamics or the coupling between the two turbulence scales. Research into the cross-scale coupling in plasmas is not limited to fusion devices. Experiment and simulation of ion and electron-scale turbulence coupling is an active area of research in both astrophysical and space plasmas 13,14. However, due to the extreme computational requirements, examples of multi-scale simulation, which captures coupling of the turbulent scales, are extremely limited 8,11,15 18 in fusion devices. All previous multi-scale work has utilized unphysical values of the electron mass ratio, 2
typically (m D /m e ) 1/2 =20.0 or 30.0, to lessen computational demands and generally was performed on model parameters (such as circular geometry and modeled values of turbulence drives). While in some cases, these approximations may yield qualitative understanding of cross-scale coupling, it has also been demonstrated to be insufficient for simulation of some plasma conditions near ITG marginal stability (a common condition in the core of tokamak plasmas) 18,andcannotbequantitativelycomparedwithexperimentalheatfluxes. In this paper we present first-of-a-kind, multi-scale simulations of a magnetically confined fusion plasma performed with all experimental input parameters and realistic electron mass ratio, (m D /m e ) 1/2 =60.0. It should be emphasized that the work presented here is not a model validation exercise. Instead, we attempt to answer the question: In realistic plasma conditions that exhibit significant ion-scale turbulence, can electron-scale turbulence drive experimentally relevant levels of electron heat flux? In the sections below these new simulations demonstrate 1.) The coexistence of ion and electron-scale turbulence in the core (r/a=0.6) of a fusion plasma 2.) enhancement of low and high-k driven heat transport by high-k turbulence dynamics and 3.) the first quantitative comparison of multi-scale simulation with experimental heat flux levels which demonstrates that electron-scale turbulence can drive experimental levels of electron heat flux in the presence of ion-scale turbulence and may explain a known discrepancy between simulation and experiment. II. EXPERIMENTAL AND SIMULATION SETUP All work in this paper focuses on a low-power Alcator C-Mod L-Mode plasma discharge that has been described previously 18 21.Weemphasizethatthisplasmapossessesnounique characteristics and represents a standard L-mode discharge operated with 1.2 MW of Ion Cyclotron Resonance Heating (ICRH), a core density of approximately 1.4 10 20 m 3,a plasma current of 0.8 MA, and a magnetic field of 5.4 T. This discharge was chosen for study based on results from previous, ion-scale simulation where agreement between simulation and experiment was identified in the ion heat and impurity particle channels while an underprediction ( 3 ) wasreportedintheelectronheatflux 19 21.Inthesepreviousworks,scans of the turbulence drive terms a/l Ti, a/l Te, a/l n, n i /n e,ande B shear were performed using ion-scale simulations (k θ ρ s < 1.3) to demonstrate that the under prediction of the electron heat flux is robust within experimental uncertainties. It was speculated that the 3
30 25 ω γ c s /a 20 15 10 5 Base Sim Base +1σ a/l Te c s /a 1.0 0.1 0 0.1 1.0 10.0 60.0 k θ ρs 0.1 1.0 10.0 k θ ρs 60.0 FIG. 1. (Color Online) The linear growth rates of modes spanning the ion and electron scales are plotted for the base case simulation parameters (red) and the base case + 1σ in a/l Te (green). discrepancy in the electron heat flux may be due to high-k TEM/ETG contributions that are not captured by standard ion-scale simulation. III. SIMULATION RESULTS All linear and nonlinear simulation work is based on massively parallel simulations performed using the GYRO code 22,23. Due to the low-beta nature of the discharge, all simulations were electrostatic. Each simulation included rotation effects ( E B rotationshear, Coriolis drift effects, and parallel flow gradients ) electron-ion collisions, debye shielding effects, Miller geometry parameterization 24,andastandard128pointvelocityspacediscretization (8 pitch angles, 8 energies, and 2 signs of velocity). Simulations included 3 gyrokinetic species: deuterium ions, electrons, and a single boron impurity species (the dominant impurity species in the experimental discharge). The final simulations presented in this paper were performed as follows: Ion-scale simulation (long wavelength only) used 8toroidalmodes,32 32ρ s simulation boxes (radial x binormal directions), 134 radial grid points, grid spacing of 0.33ρ s, and captured dynamics up to k θ ρ s = 1.4. The full multi-scale simulations (coupled short and long wavelengths) used 240 toroidal modes, 32 32ρ s simulation boxes, 1320 radial grid points, grid spacing of 2ρ e,andcapturedshort and long wavelength turbulence dynamics up to k θ ρ s =48.0 k θ ρ e =0.8. The choice of 4
maximum simulated value of k θ ρ s was chosen to effectively capture the linearly unstable modes (see Figure 1) and limited by the availability of resources. Given the extreme computational requirements of the multi-scale simulations, convergence tests were performed on the corresponding ion-scale simulations. Ion-scale simulations with a 32 32ρ s box (Q i =0.071;Q e = 0.062) can reproduce time averaged heat fluxes (MW/m 2 )obtainedfrom 112 112ρ s simulation boxes Q i =0.076;Q e =0.062). Tofurtherensureaccuracyinthe results, multi-scale simulations were run well into the saturated state ( 350 380 a/c s ). Even with optimization of the numerical settings, the multi-scale simulations pushed the limits of current computing and are amongst the most computationally expensive plasma turbulence simulations to date. They utilized approximately 18k processors and required approximately 20 Million CPU hours on the NERSC Hopper supercomputer for the completion of 2 simulations presented. The input profiles utilized are very similar to previous numerical simulation of this discharge 18,withslightchangesduetothechoiceofprofileaveragingwindow(1.0-1.4sec) and an updated analysis on the rotation profiles. The value of the a/l Ti has been modified from its experimental value, within estimated uncertainty, to match the simulated ion heat flux obtained from ion-scale simulation to the experimental value. The input values for base simulations are summarized here: r/a =0.6, a =0.22 (m), n e =1.07 (1e20), T e =1.03 (kev), ρ s =0.65 (mm), a/l n =1.01, a/l Te =3.28, a/l Ti = a/l TB =1.92, T i /T e = T B /T e =0.98, n D /n e =0.875, Z eff =1.90, q =1.56, s =1.36, κ=1.24, δ=0.11, ν ei (a/c s )=0.18, γ ExB (a/c s )=0.04, and a/c s (µs)=0.99. To understand the linear instability of ITG, TEM, and ETG in the discharge of interest, initial value, linear gyrokinetic simulations were performed spanning k θ ρ s =0.1to60.0as summarized in Figure 1. For the base simulation parameters, the linear spectra from this discharge is dominated by ITG modes (ion direction (-) and sensitive to a/l Ti variation) below k θ ρ s < 0.8. Previous ion-scale analysis of this discharge has revealed its ion heat flux-matched conditions to sit just above the nonlinear critical gradient to ITG, suggesting this plasma condition is marginally stable to low-k ITG turbulence 21. However, at higherk, (> k θ ρ s =0.8 ) electron modes, identified as TEM/ETG, become the dominant linear instability. The simple linear picture indicates that ETG remains unstable up to k θ ρ s 48.0 k θ ρ e 0.8, with only stable linear modes existing above this value. This observation motivates our choice for the maximum simulated k θ ρ s value in the multi-scale simulations. 5
30 Standard, Ion-Scale Sim kθ ρs < 1.4 25 0.045 0.030 0.015 y/ρ s 20 0.0 15 10-0.015 5-0.030 0 A -0.045-20 0-10 10 20 x/ρ s 30 Multi-Scale Simulation kθ ρs < 48.0 25 0.027 0.018 0.009 y/ρ s 20 15 0.0 10-0.009-0.018 5 B 0-0.027-20 0-10 10 20 x/ρ s FIG. 2. (Color Online) Snapshots of the potential fluctuations obtained from long wavelength, ion-scale simulation (a) and coupled short and long wavelength, multi-scale simulation (b). The coexistence of ion-scale eddies and the radially elongated ETG streamers is clearly demonstrated in the multi-scale simulation The dotted lines indicate the boundary between the physical simulation domain and buffer regions. However, to understand the coupling of the ion and electron-scale turbulence and the effects of any subdominant modes not revealed in the linear simulations, we turn to nonlinear simulation of this discharge. Snapshots of the potential fluctuations obtained from nonlinear simulation results corresponding to pure ion-scale (kθ ρs < 1.4) and coupled ion/electronscale simulation (kθ ρs < 48.0) are shown in Figure 2 in panels a and b respectively. The snapshot of the potential obtained from ion-scale simulation (Figure 2a) displays the presence of large ion-scale eddies associated with the unstable ITG turbulence in this discharge, while the coupled, multi-scale simulation (Figure 2b) demonstrates coexistence of the ion-scale eddies with clearly defined ETG streamers. The extent of these ETG driven structures is generally quite small in the θ direction with an extent in the radial direction 6
i θ s k ρ 0.02 Q i θ s k ρ 0.02 Q e Q ( k θ ρ s ) x 0.01 Ion - Scale Multi-Scale Q ( k θ ρ s ) x e 0.01 0.00 1 10 k θ ρs 0.00 48 1 10 48 k θ ρs FIG. 3. (Color Online) The heat flux spectra obtained from ion (blue) and multi-scale (red) base simulations are plotted. of 6 12ρ s. However, the visualizations of Figure 2 alone do not provide quantitative insight into the dynamics or relevance of the high-k turbulence. To better quantify the contributions and coupling of the scales, we plot the ion and electron heat flux spectra (see Figure 3) from both the ion-scale and multi-scale base simulations. We adopt a variation of the approach used by Görler and Jenko for plotting of the heat flux spectra such that the area underneath the spectra is approximately representative of the contributions to the heat flux from low and high-k 17. The ion heat flux spectra obtained from pure ion-scale simulation appears very similar to the multi-scale result, with approximately 99.9% of the total heat flux arising from contributions below k θ ρ s =1.0. In contrast, the electron heat flux spectra indicate a more complicated interaction of the ion and electron scales. The electron heat flux spectra obtained from coupled simulation displays two separate peaks: A low-k peak that occurs around k θ ρ s =0.4, that corresponds to the dominant scale for the ITG turbulence in the simulation and a high-k peak centered around k θ ρ s =6.0 k θ ρ e =0.1 thatcorresponds to ETG streamers present in the multi-scale simulation. Comparison of the heat fluxes obtained from ion-scale simulation to those obtained via coupled simulation reveals a complicated interaction of the low and high-k turbulence and is summarized in Table I. Despite the fact that all of the ion heat flux is driven at scales below k θ ρ s =1.0 forboththeionand 7
TABLE I. The ion and electron heat fluxes (MW/m 2 ) from the ion-scale and multi-scale base simulations are shown. Quantity Ion-Scale Multi-Scale Change Q i (all k θ ρ s ) 0.071 0.093 +31% Q i (k θ ρ s < 1.0) 0.071 0.093 +31% Q i (k θ ρ s > 1.0) n/a 0.000 n/a Q e (all k θ ρ s ) 0.062 0.111 +79% Q e (k θ ρ s < 1.0) 0.058 0.079 +36% Q e (k θ ρ s > 1.0) n/a 0.032 n/a multi-scale simulations, the inclusion of the high-k turbulence in the multi-scale simulation is found to increase the total ion heat flux by 31%, indicating that the electron-scale turbulence plays an important indirect role in the determination of the ion heat flux. In the electron channel, the effect of the electron-scale turbulence is significant, increasing the total driven electron heat flux by 79% above the ion-scale result. However, the distribution of this heat flux is even more intriguing as inclusion of the high-k turbulence enhances the electron heat flux driven at long wavelengths by 36% with the remainder of the electron heat flux driven at high-k. In fact, high-k contributions represent 29% of the total multiscale electron heat flux. It appears as though the role of electron-scale turbulence is both to enhance the low-k turbulence while driving electron heat flux at the high-k in the form of the ETG streamers. The enhancements found in the low-k driven ion and electron heat fluxes have important implications for comparisons of ion-scale turbulence simulation with experiment, as they suggest that quantitative comparison might be significantly altered by inclusion of electron-scale contributions. In an attempt to understand the enhancement of low-k transport in the multi-scale simulation, the low-k RMS fluctuation levels, radial correlation lengths, and phase angles of the turbulent fluctuations were evaluated from the outputs of both the ion and multi-scale simulations. A comparison of these derived simulation outputs are found in Table II. This analysis reveals increases (10 15%) in the φ, T e,andt i RMS fluctuation levels with slightly larger relative increases found in the density fluctuations ( 25%). With the exception of 8
TABLE II. The finite-n, low-k (k θ ρ s < 1.0) RMS fluctuation levels, radial correlation lengths (l r ), and phase angles (α in radians), obtained from the ion and multi-scale simulations are shown for the main ions and electrons. Quantity Ion-Scale Multi-Scale Change e φ/t e 1.11% 1.27 % +14% l r,φ (ρ s ) 8.35 8.50 +2% T e /T e 1.01 % 1.14% +13% l r,te (ρ s ) 6.49 7.26 +12% T i /T i 1.15 % 1.30% +13% l r,ti (ρ s ) 7.27 7.23 -.5% ñ e /n e.70 %.87% +24% l r,ne (ρ s ) 7.00 7.26 +4% ñ i /n i.85 % 1.08% +27% l r,ni (ρ s ) 7.67 7.46-3% α ni,t i (k θ ρ s =0.4) 1.77 1.83 +3% α ne,te (k θ ρ s =0.4) 2.53 2.53 0% α φ,ti (k θ ρ s =0.4) 1.60 1.69 +6% α φ,te (k θ ρ s =0.4) 2.49 2.49 0% a12%increaseintheradialcorrelationlengthoftheelectrontemperature,allothercorrelation lengths are effectively unchanged between the simulations. The phase angles in Table II were evaluated at k θ ρ s =0.4, the peak in the low-k ion and electron heat fluxes. Again, no significant change in the phase angles is found, suggesting that the observed low-k enhancement of the low-k heat fluxes arises almost exclusively from the combined effect of small increases in the fluctuation levels and not a shift in the phase of the fluctuations. The presented multi-scale simulation work was motivated by understanding a robust under-prediction of the experimental electron heat flux predicted by ion-scale simulation. Therefore, we wish to asses whether, in the presence of significant ion-scale turbulence, electron-scale turbulence can drive experimentally relevant levels of electron heat transport. To answer this question, we plot a comparison of the heat fluxes obtained from ion-scale and 9
i 0.03 Q i 0.03 Q e θ s k ρ Q ( k θ ρ s ) x 0.02 0.01 Base +1σ Base Sim a/l Te θ s k ρ Q ( k θ ρ s ) x e 0.02 0.01 1 10 k θ ρs 48 1 10 48 k θ ρs FIG. 4. (Color Online) The ion and electron heat flux spectra obtained from the multi-scale base case simulation and the multi-scale +1 σ simulation are plotted. multi-scale simulations with experiment in Figure 5. The base case simulations referred to below indicate simulations with no modification of the experimental turbulence drive, a/l Te. The initial ion-scale base case simulation displayed an approximately 3 under-prediction of the electron heat flux while approximately matching the mean of the experimental ion heat flux. Increases in the low and high-k driven heat flux obtained via multi-scale simulation produced ion heat fluxes still in quantitative agreement with experiment and electron heat fluxes significantly closer to matching the experimental levels. The observed (79%) increase in the total electron heat flux originating from the inclusion of electron-scale turbulence motivated an additional multi-scale simulation. Although it would be desirable to examine the sensitivity of the multi-scale results to all uncertainties in turbulence drive terms, the extreme computational requirements limited our investigation to only a single additional simulation. For an in-depth investigation of the ion-scale simulation sensitivities to experimental uncertainties, the reader is referred to Reference 21. Here, we attempt to answer a simple question: In realistic plasma conditions that exhibit significant ion-scale turbulence, can electron-scale turbulence drive experimentally relevant levels of electron heat flux? As the high-k contributions are TEM/ETG in nature, this turbulence should respond sensitively to changes in the drive term, a/l Te (Figure 1) and may have a significant impact on the nonlinear electron heat flux while leaving 10
the ion heat flux relatively unchanged. A second multi-scale simulation was completed with an increase in the ETG drive, a/l Te within its 1σ experimental uncertainty of 12.5%. A comparison of the heat flux spectra obtained from multi-scale base case and multi-scale +1σ simulation is found in Figure 4 and a direct comparison of the ion-scale base, ion-scale +1σ, multi-scale base, and multi-scale +1σ simulations with experimental ion and electron heat fluxes in found in Figure 5. The heat flux spectra obtained from both multi-scale simulations are similar in shape, with a slight downshift of the high-k electron peak to longer wavelengths (k θ ρ s 5.2) in the multi-scale +1σ simulation. The total ion heat flux obtained from this simulation remains driven almost entirely at low-k, consistent with the multi-scale base case. Relative to the ion-scale base simulation, the low-k driven electron heat flux increases by 79% in the multi-scale +1σ simulation. Ion-scale simulation with a 1σ increase a/l Te results in a slightly reduced ion heat flux (-3%) and only a slight increase in electron heat flux (+11%). Therefore, consistent with the multi-scale base case, the enhancement of the low-k electron heat flux is determined to result predominantly from the inclusion of the high-k turbulence. The electron heat flux driven at high-k is found to increase by 80% relative to the multi-scale base case and represents 36% of the total electron heat flux. Overall, relative to the ion-scale base simulation, a 162% increase in the total electron heat flux is found which moves the electron heat flux to experimental levels. When the uncertainty in the experimental and simulated heat fluxes are considered, the multi-scale electron and ion heat fluxes are now found to be in quantitative agreement with the experiment. This result suggest that electronscale turbulence may explain the robust Q e under-prediction observed in this discharge 18 21 and demonstrates for the first time that, in realistic plasma conditions with significant ion-scale turbulence, electron-scale turbulence can drive experimentally relevant levels of electron heat flux. In summary, the simulations presented here demonstrated the coexistence of ion and electron-scale turbulence in the core (r/a=0.6) of a standard tokamak plasma, indicate that low-k turbulence is enhanced by the inclusion of high-k dynamics, and demonstrate that, in the presence of significant ion-scale turbulence, electron-scale turbulence can drive experimental levels of electron heat flux. This mechanism may resolve some known discrepancies between ion-scale simulation and experiment and represents an important step towards a predictive model for fusion plasmas. 11
0.12 Q i 0.4 Q e 0.3 MW/m 2 0.08 MW/m 2 0.2 0.04 Ion-Scale Simulation Multi-Scale Simulation 1σ Experimental Uncertainty 0.1 0.00 2.5 3.0 3.5 4.0 0.0 2.5 3.0 3.5 4.0 a/l Te a/l Te FIG. 5. (Color Online) The ion and electron heat fluxes obtained from ion-scale (blue) and multiscale base (red) simulations are shown at r/a = 0.6. The shaded region is formed by 1σ uncertainties in the experimental heat fluxes and the turbulence drive a/l Te. Agreement with experiment is found if the simulation points lie within this region. Representative simulation uncertainties are plotted on the +1σ multi-scale simulation. IV. ACKNOWLEDGEMENTS The authors would like to thank the entire Alcator C-Mod team. Specifically we thank Dr. Matt Reinke for the ion temperature and rotation profiles and Dr. Tobias Görler for valuable discussion and input on multi-scale simulation. Computer simulations are part of research performed for the Center for Simulation of Plasma Microturbulence (CSPM) and were carried out at the National Energy Research Scientific Computing Center, supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. This work was also supported by DOE contract - DE-FC02-99ER54512-CMOD and in part by an appointment to the US DOE Fusion Energy Postdoctoral Research Program administered by ORISE. REFERENCES 1 W. Horton. Drift waves and transport. Review of Modern Physics, 71(3), (1999). 12
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